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Research Papers: Flows in Complex Systems

# Stokes Flow Within Networks of Flow Branches

[+] Author and Article Information
Mustapha Hellou

LGCGM (Laboratoire de Génie Civil
et Génie Mécanique),
EA3913 Univ Rennes,
INSA Rennes,
Rennes 35708, France
e-mail: mustapha.hellou@insa-rennes.fr

Franck Lominé

LGCGM (Laboratoire de Génie Civil
et Génie Mécanique),
EA3913 Univ Rennes,
INSA Rennes,
Rennes 35708, France
e-mail: franck.lomine@insa-rennes.fr

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received June 20, 2017; final manuscript received July 9, 2018; published online August 16, 2018. Assoc. Editor: M'hamed Boutaous.

J. Fluids Eng 140(12), 121110 (Aug 16, 2018) (12 pages) Paper No: FE-17-1364; doi: 10.1115/1.4040832 History: Received June 20, 2017; Revised July 09, 2018

## Abstract

Stokes flow in the branches of structured looped networks with successive identical square loops and T-junction branches is studied. Analytical expressions of the flow rate in the branches are determined for network of one, two, three, or four loops with junction head loss neglected relative to regular one. Then, a general expression of the flow rate is deduced for networks with more loops. This expression contains particularly a sequence of coefficients obeying to a recurrence formula. This sequence is a part of the fusion of Fibonacci and Tribonacci sequences. Furthermore, a general formula that expresses the quotient of flow rate in successive junction flow branches is given. The limit of this quotient for an infinite number of junction branches is found to be equal to $2+3$. When the inlet and outlet flow rates are equal, this quotient obeys to a sequence of invariant numbers whatever the ratio of flow rate in the outlet branches is. Thus, the flow rate distribution for any configuration of inlet and outlet flow rates can be calculated. This study is also performed using Hardy–Cross method and a commercial solver of Navier-Stokes equation. The analytical results are approached very well with Hardy–Cross method. The numerical resolution agrees also with analytical results. However, the difference with the numerical results becomes significant for low flow rate in the junction branches. The flow streamlines are then determined for some inlet and outlet flow rate configurations. They particularly illustrate that recirculation flow takes place in branches of low flow rate.

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## Figures

Fig. 1

Delineation of combining and dividing flow in a T-junction

Fig. 2

Evolution of the ratio of local head loss and frictional head loss as a function of Reynolds number

Fig. 3

General sketch of the flow network with M = 4, the flow direction is arbitrary drawn

Fig. 4

Sketch to explain Hardy-Cross method

Fig. 5

Sketch of the flow domain (M = 4), I=(Ia+Ib)/2

Fig. 6

Relative error theory-HC method

Fig. 7

Relative error theory-comsol

Fig. 8

Comparison of the relative error theory-HC method with relative error theory-comsol

Fig. 9

Evolution of flow rate in junction branches with β for the case of M = 4 (four loops) (a) α = 0, (b) α = 0.5, and (c) α = 1

Fig. 10

Streamlines for α = 0, β equal to 0, 0.2, 0.9, 0.99, and 1

Fig. 11

Streamlines for α = 0.5, β equal to 0, 0.2, 0.9, 0.99, and 1

Fig. 12

Streamlines for α = 1, β equal to 0, 0.2, 0.9, 0.99, and 1

Fig. 13

Pressure drop in the configuration M = 4, α = 1 and β = 0: (a) Along the lines y = D and y = D + L and (b) along the right wall of the fourth junction branch

## Errata

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